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Theorem smgrpisass 21922
Description: A semi-group is associative. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
Assertion
Ref Expression
smgrpisass  |-  ( G  e.  SemiGrp  ->  G  e.  Ass )

Proof of Theorem smgrpisass
StepHypRef Expression
1 elin 3531 . . 3  |-  ( G  e.  ( Magma  i^i  Ass ) 
<->  ( G  e.  Magma  /\  G  e.  Ass )
)
21simprbi 452 . 2  |-  ( G  e.  ( Magma  i^i  Ass )  ->  G  e.  Ass )
3 df-sgr 21920 . 2  |-  SemiGrp  =  (
Magma  i^i  Ass )
42, 3eleq2s 2529 1  |-  ( G  e.  SemiGrp  ->  G  e.  Ass )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726    i^i cin 3320   Asscass 21901   Magmacmagm 21907   SemiGrpcsem 21919
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-in 3328  df-sgr 21920
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